Guest Lecture for ECON 3100
October 15, 2024
\[U_i = \sum_{t=T_0}^{T} \delta^{t-T_0} \left( d_{ist} \cdot \omega_{ist} + (1 - d_{ist}) \cdot \eta_{ist} \right)\]
\(U_i\) = utility of individual \(i\) over the periods from \(t\) to \(T\)
\(\delta\) = discount factor; reflects time preferences; \(0 < \delta < 1\),
\(d_{ist} \in \{0, 1\}\); whether employed in sports
\(\omega_{ist}\) = sport wage
\(\eta_{ist}\) = non-sport wage
\[\eta_{ist} = \mu_i - \theta_{ist}\]
\(\mu\): worker’s human capital value
\(\theta\): reduction in working years due to physical toll from violent sports
\[\theta_{ist} = \gamma_{st} + \epsilon_{ist}\] - \(\gamma\) = known (“inherent”) dangers of the sport
\[\omega_{ist} = \rho_{st} \cdot \upsilon_{ist} + \beta_{ist} \cdot \theta_{ist}\]
\(\rho\) = market demand
\(\upsilon\) = value of the marginal product of labor (relative to replacement worker)
\(\beta\) = workers’ bargaining power
\[\beta_{ist} = \mu_i + \pi_{st} \cdot \upsilon_{ist}\] - \(\pi\) = strength of collective bargaining agreement
\[ \Delta_{it} (\upsilon_{ist} + \epsilon_{ist} \cdot (\upsilon_{ist} + 1)) \geq 0 \]
\[\Delta_{it}( \upsilon_{ist}) \geq 0\]
\[ \Delta_{it} (\upsilon_{ist} + \epsilon_{ist} \cdot (\upsilon_{ist} + 1)) \geq 0 \]
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |||
|---|---|---|---|---|
| Model: | (1) | (2) | (3) | |
| Panel A (Time as Teammates) | ||||
| CTE (Post) x Teammate | 2.53*** | 4.30*** | 4.67*** | |
| (0.80) | (1.22) | (1.69) | ||
| Pre-Treatment Mean | 29.03 | 25.69 | 24.87 | |
| % Change | 0.09 | 0.17 | 0.19 | |
| Observations | 229,833 | 218,853 | 215,226 | |
| Mean Pre-Treatment Age | 27.75 | 28.59 | 29.31 | |
| Length of Time Teammates (\(L\)): | \(L \geq 1\) | \(L \geq 2\) | \(L \geq 3\) | |
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |||
|---|---|---|---|---|
| Model: | (1) | (2) | (3) | |
| Panel A (Recency of Treatment) | ||||
| CTE (Post) x Teammate | 3.14** | 1.78* | 1.91** | |
| (1.45) | (0.96) | (0.88) | ||
| Pre-Treatment Mean | 10.45 | 22.27 | 28.29 | |
| % Change | 0.3 | 0.08 | 0.07 | |
| Observations | 95,049 | 202,824 | 219,366 | |
| Recency of Treatment (D): | \(D \leq 2\) | \(D \leq 4\) | \(D \leq 6\) | |
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |||
|---|---|---|---|---|
| Model: | (1) | (2) | (3) | |
| Panel B (Recency & 3+ Years Teammates) | ||||
| CTE (Post) x Teammate | 7.22* | 4.62* | 3.81* | |
| (4.05) | (2.65) | (2.29) | ||
| Pre-Treatment Mean | 10.45 | 22.27 | 28.29 | |
| % Change | 0.69 | 0.21 | 0.13 | |
| Observations | 39,231 | 68,256 | 73,035 | |
| Recency of Treatment (D): | \(D \leq 2\) | \(D \leq 4\) | \(D \leq 6\) | |
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |
|---|---|---|
| Model: | (1) | (2) |
| Car Accident (Post) x Teammate | -0.01 | |
| (0.01) | ||
| Disease (Post) x Teammate | 0.00 | |
| (0.01) | ||
| Mean of Dep. | 0.19 | 0.19 |
| % Change | -0.06 | 0.00 |
| Observations | 95,533 | 70,891 |
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |||
|---|---|---|---|---|
| Sample: | Full | Hockey | Football | |
| Model: | (1) | (2) | (3) | (4) |
| CTE (Post) x Teammate | 3.59** | 2.48* | 1.53 | 4.08** |
| (1.45) | (1.46) | (1.41) | (1.88) | |
| \(sinh^{-1}\)(Opp. Cost) | -0.48*** | -0.06*** | -0.09*** | -0.02 |
| (0.01) | (0.01) | (0.01) | (0.02) | |
| CTE (Post) x Teammate x \(sinh^{-1}\)(Opp. Cost) | -0.37*** | -0.32*** | -0.06 | -0.52*** |
| (0.09) | (0.09) | (0.08) | (0.12) | |
| Controls: Ct. Total & Length | \(\checkmark\) | \(\checkmark\) | \(\checkmark\) | |
| Observations | 131,424 | 131,424 | 18,679 | 112,745 |
“Let us suppose that …[a great, distant empire], with all its myriads of inhabitants, was suddenly swallowed up by an earthquake, and let us consider how a man of humanity […], who had no sort of connection with that part of the world, would be affected upon receiving intelligence of this dreadful calamity. He would, I imagine, first of all, express very strongly his sorrow for the misfortune of that unhappy people, he would make many melancholy reflections upon the precariousness of human life, and the vanity of all the labours of man, which could thus be annihilated in a moment…
And when all this fine philosophy was over, when all these humane sentiments had been once fairly expressed, he would pursue his business or his pleasure, take his repose or his diversion, with the same ease and tranquillity, as if no such accident had happened. The most frivolous disaster which could befall himself would occasion a more real disturbance.
If he was to lose his little finger to-morrow, he would not sleep to-night; but, provided he never saw them, he will snore with the most profound security over the ruin of a hundred millions of his brethren, and the destruction of that immense multitude seems plainly an object less interesting to him, than this paltry misfortune of his own.”
- Adam Smith, The Theory of Moral Sentiments
Unit fixed-effects: \(\Delta X_i = X_{it} - \frac{1}{T} \sum_{\tau = 1}^T x_{i \tau}\)
Time fixed-effects: \(\Delta X_i = X_{it} - \frac{1}{I} \sum_{\iota = 1}^I x_{\iota t}\)
For notational simplicity, I denote unit and time fixed-effects as \(\Delta_i\) and \(\Delta_t\) respectively.
All athletes diagnosed with CTE from 2005-2008 shared numerous common traits: long professional careers, disproportionate exposure to repeated head trauma, and violent styles of play.
This changed in 2009 with the discovery of CTE in Chris Henry.
In 14 months in 2010-2011, 4 hockey players (3 of whom were still active professionals) died and were diagnosed with CTE at the ages of 45, 35, 28, and 27. Back
| Player | Career | Date of Death | Teammates at τ = -4 |
|---|---|---|---|
| Justin Strzelczyk | 1990-1998 | 9/30/2004 | 61 |
| Shane Dronett | 1992-2001 | 1/21/2009 | 40 |
| Chris Henry | 2005-2009 | 12/17/2009 | 137 |
| Junior Seau | 1990-2012 | 5/2/2012 | 157 |
| Jovan Belcher | 2009-2012 | 12/1/2012 | 116 |
| Paul Oliver | 2008-2011 | 9/24/2013 | 115 |
| Adrian Robinson | 2012-2013 | 5/16/2015 | 115 |
| Player | Career | Date of Death | Teammates at τ = -4 |
|---|---|---|---|
| Tyler Sash | 2011-2012 | 9/8/2015 | 80 |
| Aaron Hernandez | 2010-2012 | 4/19/2017 | 71 |
| Daniel Te’o-Nesheim | 2010-2013 | 10/29/2017 | 117 |
| George Atkinson | 2014-2016 | 12/2/2019 | 117 |
| Max Tuerk | 2017-2017 | 6/20/2020 | 68 |
| Vincent Jackson | 2005-2016 | 2/15/2021 | 92 |
| Phillip Adams | 2010-2015 | 4/8/2021 | 114 |
| Demaryius Thomas | 2010-2019 | 12/9/2021 | 199 |
| Player | Career | Date of Death | Teammates at τ = -4 |
|---|---|---|---|
| Bob Probert | 1986-2002 | 7/5/2010 | 53 |
| Derek Boogaard | 2006-2011 | 5/13/2011 | 102 |
| Rick Rypien | 2006-2011 | 8/15/2011 | 92 |
| Wade Belak | 1997-2011 | 8/31/2011 | 151 |
| Steve Montador | 2002-2012 | 2/15/2015 | 141 |
| Marek Svatoš | 2004-2011 | 11/4/2016 | 59 |
| Dependent Variable: | \(\mathbb{1}\)(Exit) * 100 | |||
|---|---|---|---|---|
| Model: | (1) | (2) | (3) | |
| Panel B (Age) | ||||
| CTE (Post) x Teammate | 0.10 | 3.70** | 0.56 | |
| (2.22) | (1.53) | (1.72) | ||
| Pre-Treatment Mean | 10.8 | 17.31 | 36.27 | |
| % Change | 0.01 | 0.21 | 0.02 | |
| Observations | 97,794 | 131,022 | 32,643 | |
| Pre-Treatment Mean Age | 22.36 | 26.03 | 32.25 | |
| Age Percentile (\(P\)): | \(P < \frac{1}{3}\) | \(\frac{1}{3} \leq P < \frac{2}{3}\) | \(P \geq \frac{2}{3}\) | |